Arithmetic, Geometry, Cryptography and Coding Theory: 17th International Conference Arithmetic, Geometry, Cryptography and Coding Theory June 10-14, ... France (Contemporary Mathematics, 770)

✍ Scribed by Stephane Ballet (editor), Gaetan Bisson (editor), Irene Bouw (editor)

• Publisher: AMS
• Year: 2021
• Tongue: English
• Leaves: 322
• Category: Library

✦ Synopsis

From a June 2019 international conference in Marseilles, 15 original research papers (two in French) connect arithmetic and algebraic geometry. Their topics include the number of effective divisors in algebraic function fields defined over a finite field, towards good families of codes from towers of surfaces, Sato-Tate groups of abelian threefolds: a preview of the classification, stable models of plane quartics with hyper-elliptic reduction, and the quadratic hull of a code and the geometric view on multiplication algorithms. Annotation ©2021 Ringgold, Inc., Portland, OR (protoview.com)

✦ Table of Contents

Cover
Title page
Contents
Preface
A new upper bound for the largest complete \boldmath(𝑘,𝑛)-arc in \boldmath\PG(2,𝑞)
1. Introduction
2. Some basic equations
3. Non-existence of some \boldmath(𝑘,𝑛)-arcs in \boldmath\PG(2,𝑞)
4. New largest bound
5. Application of Theorem 4.1
Acknowledgment
References
Bounds on the minimum distance of algebraic geometry codes defined over some families of surfaces
1. Introduction
2. Background
3. The minimum distance of codes over some families of algebraic surfaces
4. Four improvements
5. An example: surfaces in \boldmathℙ³
Acknowledgments.
References
On the number of effective divisors in algebraic function fields defined over a finite field
1. Introduction
2. Contents
3. Non-asymptotical case
4. Asymptotical case
Appendix A.
References
The absolute discriminant of the endomorphism ring of most reductions of a non-CM elliptic curve is close to maximal
1. Introduction
2. Proof of the main theorem
Acknowledgments
References
Toward good families of codes from towers of surfaces
Introduction
1. Codes from surfaces
2. Infinite étale towers of surfaces
3. Open problems
Acknowledgments
References
Appendix A. On the étale site of marked schemes, \normalfont by Alexander Schmidt
Acknowledgment
References
Sato–Tate groups of abelian threefolds: a preview of the classification
1. Introduction
2. Background on Sato–Tate groups
3. Classification: an overview
4. Realization
References
Arithmetic, Geometry, and Coding Theory: Homage to Gilles Lachaud
1. Gilles Lachaud’s early works
2. Gilles Lachaud: friend and mathematician
3. A Tribute to Gilles Lachaud
4. Gilles Lachaud’s work
References
Elliptic curves with large Tate–Shafarevich groups over \boldmath𝔽_{𝕢}(𝕥)
Introduction
1. Invariants of elliptic curves over function fields
2. The elliptic curves \boldmath𝐸_{𝛾,𝑎}
3. Preliminaries on character sums
4. The \boldmath𝐿-function of \boldmath𝐸_{𝛾,𝑎}
5. Non-vanishing of \boldmath𝐿(𝐸_{𝛾,𝑎},𝑇) at the central point and consequences
6. Distribution of the Kloosterman sums \Kloosnᵧ(𝑣)
7. Estimates of the central value 𝐿(𝐸_{𝛾,𝑎},𝑞⁻¹)
8. Proof of Theorem C
Acknowledgments
References
On Sato–Tate distributions, extremal traces, and real multiplication in genus 2
1. Introduction
2. Background and notation
3. Weyl integration formula
4. Taylor expansions for the trace function
5. On results of Lachaud
6. Distribution of extremal traces in families
7. Conclusion
Acknowledgments
References
La trace et le deltoïde de \boldmath\SU(3)
1. Introduction
2. Le groupe unitaire
3. Le deltoïde
4. Loi du caractère standard
5. Loi de la longueur de la trace
6. Loi de la norme de la trace
7. Loi du caractère de la représentation adjointe
8. Loi de la partie réelle du caractère standard
9. Loi de la partie imaginaire du caractère standard
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Stable models of plane quartics with hyperelliptic reduction
1. Introduction and main result
2. Link with theta constants
3. A Riemann model providing a good \PHM
4. Bitangents of a smooth plane quartic
5. The algorithm and an example
Acknowledgment
References
Courbes de genre \boldmath3 avec \boldmath𝑆₃ comme groupe d’automorphismes
1. Introduction
2. Caractéristique différente de 2 et 3
3. Le cas de caractéristique 3
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Bornes sur le nombre de points rationnels des courbes : en quête d’uniformité
1. Introduction
2. Comptage euclidien
3. Variante de Lang-Silverman
Appendice A. Corrigendum à “Minorations des hauteurs normalisées des sous-variétés de variétés abéliennes II” par Sinnou David et Patrice Philippon
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The quadratic hull of a code and the geometric view on multiplication algorithms
1. Introduction
2. Multiplication reductions
3. An example in dimension 2
4. The canonical point
5. The quadratic hull
6. Application to geometric realizations
7. Experimental results
8. Further properties of the quadratic hull
Acknowledgments
References
Serre’s genus fifty example
1. Introduction
2. The elliptic curve
3. Constructing the curve via Artin-Schreier extensions
4. More equations and intermediate curves
Acknowledgment
References
Back Cover